`1/20+1/30+1/42+1/56+1/72+1/90+1/110+1/132` is equal to

Options

A. `1/8`
B. `1/7`
C. `1/6`
D. `1/10`

Answer

To solve the expression \( \frac{1}{20} + \frac{1}{30} + \frac{1}{42} + \frac{1}{56} + \frac{1}{72} + \frac{1}{90} + \frac{1}{110} + \frac{1}{132} \), we can simplify each fraction using a common technique. Let's break it down step by step.

### Step 1: Rewrite each fraction
We can express each fraction in terms of consecutive integers:

- \( \frac{1}{20} = \frac{1}{4 \cdot 5} = \frac{1}{4} - \frac{1}{5} \)
- \( \frac{1}{30} = \frac{1}{5 \cdot 6} = \frac{1}{5} - \frac{1}{6} \)
- \( \frac{1}{42} = \frac{1}{6 \cdot 7} = \frac{1}{6} - \frac{1}{7} \)
- \( \frac{1}{56} = \frac{1}{7 \cdot 8} = \frac{1}{7} - \frac{1}{8} \)
- \( \frac{1}{72} = \frac{1}{8 \cdot 9} = \frac{1}{8} - \frac{1}{9} \)
- \( \frac{1}{90} = \frac{1}{9 \cdot 10} = \frac{1}{9} - \frac{1}{10} \)
- \( \frac{1}{110} = \frac{1}{10 \cdot 11} = \frac{1}{10} - \frac{1}{11} \)
- \( \frac{1}{132} = \frac{1}{11 \cdot 12} = \frac{1}{11} - \frac{1}{12} \)

### Step 2: Substitute back into the expression
Now we can substitute these back into the original expression:

\[ \left( \frac{1}{4} - \frac{1}{5} \right) + \left( \frac{1}{5} - \frac{1}{6} \right) + \left( \frac{1}{6} - \frac{1}{7} \right) + \left( \frac{1}{7} - \frac{1}{8} \right) + \left( \frac{1}{8} - \frac{1}{9} \right) + \left( \frac{1}{9} - \frac{1}{10} \right) + \left( \frac{1}{10} - \frac{1}{11} \right) + \left( \frac{1}{11} - \frac{1}{12} \right) \]

### Step 3: Simplify the expression
Notice that many terms will cancel out:

\[ \frac{1}{4} - \frac{1}{12} \]

### Step 4: Find a common denominator
To subtract \( \frac{1}{4} \) and \( \frac{1}{12} \), we need a common denominator. The least common multiple of 4 and 12 is 12.

- Convert \( \frac{1}{4} \) to have a denominator of 12:
\[ \frac{1}{4} = \frac{3}{12} \]

### Step 5: Perform the subtraction
Now we can perform the subtraction:

\[ \frac{3}{12} - \frac{1}{12} = \frac{2}{12} \]

### Step 6: Simplify the fraction
Now simplify \( \frac{2}{12} \):

\[ \frac{2}{12} = \frac{1}{6} \]

### Final Answer
Thus, the value of the expression \( \frac{1}{20} + \frac{1}{30} + \frac{1}{42} + \frac{1}{56} + \frac{1}{72} + \frac{1}{90} + \frac{1}{110} + \frac{1}{132} \) is:

\[ \frac{1}{6} \]